When a wave travel in a medium, the particle displacement is given by the equation $\mathrm{y}=a \sin 2\mathrm{\pi } (\mathrm{bt}-\mathrm{cx})$ where a, b and c are constants. The maximum particle velocity will be twice the wave velocity if

Equation of the harmonic progressive wave given by $y=a \sin 2\mathrm{\pi } (\mathrm{bt}-\mathrm{cx})$.

Here $\mathcal{v}$ = b 

            $k=\frac{2\mathrm{\pi }}{\lambda }=2\mathrm{\pi c}$take, $\frac{1}{\lambda }=c$

       $\therefore$ Velocity of the wave =$\mathcal{v}\lambda =b\frac{1}{c}=\frac{b}{c}$

   $\frac{dy}{dt}=a2\mathrm{\pi b} \cos 2\mathrm{\pi } (\mathrm{bt}-\mathrm{cx}) =\mathrm{a\omega } \cos (\mathrm{\omega t}-\mathrm{kx})$   

Maximum particle velocity  $=a\omega =a2\pi b=2\pi ab$

 given this is $2\times \frac{b}{c}$ i.e. $2\mathrm{\pi a}=\frac{2}{\mathrm{c}} \mathrm{or} \mathrm{c}=\frac{1}{\mathrm{\pi a}}$